The Stone--and other Paradoxes of Infinity

[Antone]

Paradox is an imagined product of consciousness resulting from a state of ignorance that has yet to perceive that all is within. The function of paradox is that a reality is allowed to be created that successfully bridges eternity while providing incentive for consciousness to investigate and experiment; not unlike a new preschooler learning/manipulating their surrounding reality for the purpose of learning lessons.

Excellent insight soulsurviver, this resonates well with my present understanding.

I believe that one of the best explanations of [what a paradox is] and [what it is good for is] is given by Rebecca Goldstein, who said:

Paradoxes have often been found lurking about in the deepest palces of thought. Their presence is often the signal (like the canary dying?) that we have managed, sometimes unwittingly, to stumble on a deep and problematic place, a fissure in the foundations...

Adam Morton expands on this way of thinking with the following: Suppose that there were some profound flaw in the way human beings think. This would spoil all our reasoning; but since it would also spoil the reasoning we use to check our reasoning, we would never know that our thinking had gone wrong.

I agree with them both, and believe that "real" paradoxes are the indication that something is wrong with the way we understand reality. Often times, it is a fairly small misunderstanding, but as Stanley P. Gudder has said, A slight variation in the axioms at the foundations of a theory can result in huge changes at the frontier. So it doesn't take much being out of wack to create a paradox.

One of the interesting things about philosophy is that many of the attempt to fix a problem or paradox tends to succeed with respect with the problem they are addressing. But they create new problems elsewhere. To me, this is the surest sign that we are dealing with a real paradox.

[Antone]

But you said the stone is never infinite, at any given instant it is finite. So God never creates an infinitely large stone. The question is "Can God (or anyone) create an infinite object". The answer is no because there is no such "thing" as 'an' infinite object. You said as much yourself. The question itself is nonsensical.

The way I see it, what makes the stone infinite is the fact that it is BEING created--and it will continue to be so FOREVER. What makes this possible (and saves the question from being nonsensical) is that the God is IMMORTAL, and so will live forever.

Obviously, it is a make-believe question. But I belive the logic I use is sound.

Also, keep in mind that the actual question doesn't say anything at all about creating or lifting an infinitely large stone. That was a term that I introduced in my logical argument to say that God could: create a stone so largel that he couldn't lift it--(without the situation being paradoxical). That last bit about not being paradoxical is my addition, but it is in keeping with the spirit of the question, I think, if not the letter of it.

[altonhare]

Antone,

Are these different:

An infinite/endless/unbounded object

An incessantly growing object

[Antone]

There is nothing logical about square-circles. One cannot "imagine" such a concept ... Can [an omnipotent] being create a square circle? It doesnt matter how one asks ,if this is what they actually mean then the question is invalid ...

Interesting point, but I think you miss the distinction of [what exactly a concept is]--at least as I understand them. A concept is the sort of thing that doesn't have to be producible in physical reality.

Consider, for instance, my [concept of a unicorn]. The most important part of my understanding of a [unicorn] is that it is a mythical creature with one horn. Now, the definition of mythical is imaginary, fictitious. So the definition of a [unicorn] is basically a one horned creature that doesn't exist. So an omnipotent being can NOT create a unicorn because they canot bring into existence a [creature that is defined by its own lack of existence]. Such a task is paradoxical, by definiton.

Now, as I recall, my comment about the square-circle was not that it was logical, but rather that it was a concept that probably didn't have a physical counter-part. The unicorn is another. But out of a desire to be rediculous let me suggest a few possibility of what a square-circle might be. One possibility is the [juxtuposition of the Riemann sphere on the cartesian graph.] We can draw a line from the topmost point of the Riemann sphere to any point on the graph--thus creating a sort of theoretical square out of a sperical structure, so to speak.Point is, the first time I saw these structures together one of the first things I thought of was that this made a decent argument for being a viable [square circle] concept.

Keeping it simpler--a sphere is not an entirely unreasonable shape to give that name, because it is a circle that has been squared. Okay, perhaps that one is a bit of a stretch--but a Seahorse isn't really a HORSE either, so sue me. lol.

Or we could work the other way, and take a cube that is inside of a sphere, such that each corner of the cube is touching the spere--then imagine warping the lines of the cube outwards until they are toucing the surface of the circle. Again a bit of a stretch--but the point is that there are lots of possibilities of what such an imaginary thing might be. Because it is imaginary and because it is so thouroughly undefined--I can create any idea I want and call it a [square circle]. And my version of that concept has as much right to be called a [square circle] as anyone elses. Your conept of the [square circle] may simply be [a couple of words thrown together that don't have any meaning]. And again that's as valid as anything anyone else comes up with.

The fact that we come up with different concepts for the same term shouldn't be problematic either. The same thing happens all the time for every word we use. Sometimes the variations in conceptual meaning are infinitesimally small. Other times they are quite significant. Altonhaire and I (for example) give quite different conceptual meanings to all sorts of words--including words as common as: concepts, objects, things, points, and so forth. In my experience, I have observed that our conceputal versions of these words are so radically different that they do not even remotely resemble each other in many cases.

[altonhare]

Or we could work the other way, and take a cube that is inside of a sphere, such that each corner of the cube is touching the spere--then imagine warping the lines of the cube outwards until they are toucing the surface of the circle.

But a cube warped in such a way is no longer a cube by definition. It's just another sphere. A cube is *defined* as having uniform extent in three mutually perpendicular directions. As soon as you warp it, even the tiniest bit, it ceases being a cube *by definition*.

One possibility is the [juxtuposition of the Riemann sphere on the cartesian graph.]

Probably a bad idea to bring complex numbers into these discussions in general, it'll probably tend to result in tangents, which I will here try to avoid.

So, which object is the "square circle"? Is it the sphere you started with, the shape produced on the graph by mapping parts of the sphere onto it, or something else?

Probably bad to bring in Riemann shapes in general. For instance the Riemann "triangle" is a misnomer because it is not a triangle, again by definition. It's just a certain shape that you point to. The Riemann sphere is also not a sphere, by definition.

Because it is imaginary and because it is so thouroughly undefined--I can create any idea I want and call it a [square circle].

Right, you can point at something and call it a square circle if you want. But you cannot present an object, even in your imagination, that fulfills contradictory criteria. In the standard geometry definitions of squares and circles there is no object, not even one you can imagine, that fulfills both sets of criteria.

You are missing the whole point of the square-circle argument. The point of mentioning square circles is that you cannot present or even imagine an object or concept that fulfills contradictory criteria.

Let's look at the "stone paradox".

Define: God can do anything.

Ask: Can God...

Answer: Yes.

Done.

But this doesn't satisfy people for some reason. They want to put something to the right of "God...". The question, then, has nothing to do with God because we already established that he can do anything. The question becomes, what is to the right of the word "God"?

I can barely bring myself to pursue this line of thought and reasoning, I may as well sit down with Alice and the Mad Hatter.

God: Can do anything

Pick up: Something God does

Stone: Something God can't "pick up".

So, you ask, can God create a stone he can't pick up? The answer is that this question is invalid, it violates our own definition of "God"! We just_got_done_saying he can "do anything". Then we define "stone" as something he can't pick up! Which is it??? Can God do anything or not?

So, you see, people talk around in circles and whittle away countless hours because they insist on contradicting themselves, and then wonder why they can't seem to come to a resolution.

Antone essentially changed the question. Instead of defining the stone as something God can't pick up, he defined the stone as incessantly growing. Then this problem is *trivial*:

God: Can do anything

Can God pick up an incessantly growing stone?

Answer: Yes, God can do anything.

Done.

Person A: God can do anything

Person B: Can God...

Person A: YES! Jeez shut up about it already, I just said he could do ANYTHING.

Person B (smug): But can he create a stone he CANNOT pick up!

Person A: You're the one trying to say there's a stone he cannot pick up. Do you disagree that he can do anything?

Person B: I'm saying he cannot create a stone he cannot pick up.

Person A: But I defined God as able to do anything. You're defining it differently.

Person B: No, I'm using your definition, God can create a stone he cannot pick up, but then he cannot pick it up! Either that, or he cannot create the stone in the first place!

Person A: *shoots person B*

Person A asks God for forgiveness and goes to Heaven.

[Antone]

Can a paradox exist? Can identity be violated? ... CAN a thing be what it is and what it is not? Can a paradox exist. Yes or NO?

That depends on how you define a paradox. And there is a lot of controversy on this.

If the fact that an [omnipotent being can't create a unicorn] is paradoxical, then yes, a paradox can exist. There are many rules of nature that not only appear to us as contradictory--they are contradictory, at least at some level. Subatomic particles express a wave and a particle aspect. That's a paradox.

I believe that paradox, in this last sense, is the most fundamental aspect of all reality. Any structure that carries information is necessarily paradoxical in the sense that it [is] and [is not] in some way.

[Apple A] is the same as [Apple B] and yet it is also different at the same time. This doesn't represent a true paradox, because they aren't the [same and different] in the same way at the same time. They are the same in the sense that they are both [instances of the concept apple]; and they are different in the sense that they are [different physical objects].

If we don't concern ourselves with cases like the unicorn (which are based on violations of a definition) and we don't consider situations of this sort of reciprocal overlap to be considered paradoxical--then I believe that there aren't any true paradoxes. Everything that we consider to be paradoxical can be adequately explained by defining things in an appropriate way. (This assumes that there aren't any other trivial classes of paradox that I haven't forgot to include, of course.)

[Antone]

Are these different:
An infinite/endless/unbounded object
An incessantly growing object

The way I interpret/percieve things [unbounded] tends to refer to the [spatial aspect of infinity] that I mentioned in previous posts. So an [unbounded object] would be different from an [incessantly growing object] in much the same way that the [wave aspect] of a subatomic particle is different from the [particle aspect].

But I think it would be accurate to say that
An incessantly growing object
An infinitely growing object
An endlessly growing object
are equivalent statements; while
An unbounded object
An infinite object
An endless object
are also equivalent statements.

[altonhare]

Antone:

Great, you agree that an object that is growing and growing ad nauseum is different than 'an' unbounded "object". In the former case we are referring to something happening, something dynamic, fundamentally motion. We make this clear by using a verb (growing) and an adverb (incessantly). In the latter case we are referring to something static, an object's extent at some instant. We make this clear by using a noun (object) and an adjective (unbounded).

Can you imagine 'an' unbounded "object"? Not even discussing if it can exist or not. I can imagine things that may not exist. Can you even *imagine* an endless/unbounded object?

[Antone]

which object is the "square circle"? Is it the sphere you started with, the shape produced on the graph by mapping parts of the sphere onto it...

Neither. It is the union of both, much as [batchlor] is the union of [unmarried] and [male].

you can point at something and call it a square/circle if you want. But you cannot present an object, even in your imagination, that fulfills contradictory criteria.

Of course not, but I can imagine the possibility that there might be such a thing. And that is the concept of the [square cirlce]. Which we can make any number of specific attempts to satisfy with real examples--none of which can be entirely satisfactory, by defintion.

But we've already expressed our different opinions elsewhere on this, so lets let it go at that, if you don't mind.

The question, then, has nothing to do with God because we already established that he can do anything.[

I disagree. I've already mentioned [physically creating a mythical creature] as being something that an omnipotent being cannot--because the defintion of a mythical creature prevents it from being physical. Logically, there are things that we can ask of God that God cannot do. But this does not happen to be one of them, according to my argument.

The answer is that this question is invalid, it violates our own definition of "God"!

There are riddles for which this might be a valid answer. So I won't deny that this is one possible way to understand things. But again, I believe that my way of understanding things gives us a way to make the question one that we can meaningfully answer in another way.

Antone essentially changed the question. Instead of defining the stone as something God can't pick up, he defined the stone as incessantly growing.

I think you need to reread the actual argument. Your summation of it is not only grossly oversimplified--but technically inaccurate. The point was that God cannot pick up the [infinitely large stone] not that he could pick it up.

The wording of the riddle is important. I suspect that with a little effort it could be reword it so that the answer was no.

[Antone]

Can you imagine 'an' unbounded "object"?

Of course I can.

The unbounded object is the [endlessly growing object] condensed to the [now]. It is all the sizes that the growing boulder [has or will take] over the course of its infinite existence. That's why it has no boundaries--because it is all the sizes and all the shapes that the boulder [has or will ever] be. It is boundaryless because it is spatially vague--and no matter how large a size of boulder you can imagine, the unbounded boulder will include a size that is larger still, becasue it includes all the increasing sizes that will ever be. And again, there will be no end to those sizes.

Such an unbounded object cannot be actualized in physical reality. It is strictly a concept.

But to some extent or another, all concepts can be interpreted as unbounded objects. The concept [apple] includes all of the possible shapes that an apple can take. In other words, all of the shapes that an apple [ever has or ever will] take. That's why the [concept apple] is phsycally vague.

[altonhare]

Of course not, but I can imagine the possibility that there might be such a thing.

lol.

And that is the concept of the [square cirlce].

But what use is a concept that nobody can even imagine? What good is it to science and physics, where we are explaining what happens in nature? What value is 'an' unimaginable concept?

So what if you can sit around and utter the words "I imagine X might possibly exist" and name that activity "square circle"? What does the name you apply to your solo monologues have to do with anthing?

I disagree. I've already mentioned [physically creating a mythical creature] as being something that an omnipotent being cannot--because the defintion of a mythical creature prevents it from being physical. Logically, there are things that we can ask of God that God cannot do. But this does not happen to be one of them, according to my argument.

Wrong. The word "create" means "bringing into existence". So this is the "logic" you've set up:

Create: To bring into existence

Unicorn: *DEFINED* as nonexistent

Question: Can a unicorn be created?

Translated: Can an object that is defined as nonexistent exist?

Silliness. The question has nothing to do with God at all. If the unicorn is created it could exist and the assumption (unicorn is nonexistent) is wrong. If the unicorn is not created it's because it does not exist and cannot exist, and the definition is correct. If God is *defined* as able to "do anything" then the answer to the question:

Can God...

Is answered "Yes". The only way people get into arguments about this is someone tries to change the definition(s) and re-ask the question, like I showed above. A person tries to define something God cannot do, when we just said he could do anything. As soon as I define God as able to do anything, the discussion is over. He can do anything, he's defined that way. Please stop trying to redefine it other person.

There are riddles for which this might be a valid answer. So I won't deny that this is one possible way to understand things. But again, I believe that my way of understanding things gives us a way to make the question one that we can meaningfully answer in another way.

You're in the habit of seeking answers to invalid questions? No wonder we have so much trouble!

I think you need to reread the actual argument. Your summation of it is not only grossly oversimplified--but technically inaccurate. The point was that God cannot pick up the [infinitely large stone] not that he could pick it up.

The problem is there is no such thing as 'a' infinitely large stone. At any given moment God can pick it up the incessantly growing stone. At any given moment the stone is finite, no matter how long God waits. It doesn't matter that God is defined as immortal. No matter how long he waits it is still finite. The question is invalid, the stone is never infinite. 'A' infinite stone is a self-contradiction.

If people would stop pursuing self-contradictory lines of reasoning they could save a lot of time and effort.

[altonhare]

Can you imagine 'an' unbounded "object"?

Of course I can.

Show it to me! Any example, any at all! Any format!

[altonhare]

The unbounded object is the [endlessly growing object] condensed to the [now].

But now, at this instant, it's bounded! You said so yourself, at any instant the stone/object is finite. Follow your own logic.

[altonhare]

Okay class, your final exam for logic 101 is a single question and you may talk amongst yourselves.

Premise: God can do anything

Question: Can God do A.

Don't expect the results till next week, I'm getting hammered, hope I don't see you all again.

[altonhare]

As soon as you try to push the question past "Can God..." with the answer "yes" the question no longer has anything to do with God, it has everything to do with if what comes after ... contradicts your own definition of God.

Premise: God can do anything.

Question: Can God create something that doesn't exist?

Answer: Yes, God can do anything.

Premise: There is a stone that God cannot pick up.

Question: Can God create this stone?

By asking this question you have implicitly and retroactively redefined the word "God". You have placed a restriction on God's power, stating that he is no longer omnipotent. You have redefined God as "can do anything, except pick up this stone". But if you claim he can pick up the stone you implicitly and retroactively redefine the word "stone" as "can be picked up by God". If you claim he can't pick up the stone you've implicitly and retroactively redefined the word "God" as "can do anything, except pick up this stone".

Premise: This stone/unicorn/whatever cannot exist.

Question: Can God create something that cannot exist?

By asking this question you have implicitly and retroactively redefined the word "God". You have placed a restriction on God's power, saying that there is a stone that he cannot create. God is now "can do anything, except create this stone". But if you claim God creates the stone you implicitly and retroactively redefine the word "stone". Stone is now "can exist" instead of "cannot exist". If you claim god cannot create the stone, again you have implicitly and retroactively redefined the word "God" as "can do anything, except create this stone".

You can run around in circles for the rest of your life like this! The only way to answer a question is to define your words one way and then use them that way. It's like looking at an index card that says "read this, then turn over" and never stopping!



Are we done here?